Geometric phases of light: insights from fibre bundle theory

Abstract

Geometric phases are ubiquitous in physics; they act as memories of the transformation of a physical system. In optics, the most prominent examples are the Pancharatnam-Berry phase and the spin-redirection phase. Additional geometric phases of light have recently been discovered, following technological advances in phase and polarization structuring. Fibre bundle theory shows how a state transformation leads to a geometric phase. Despite its fundamental importance, this framework has been overlooked in optics. In this colloquium, we interpret geometric phases of light in terms of fibre bundles. We show that fibre bundle theory does not only sheds light on the origin of geometric phases of light, but also lays the foundations for the exploration of high dimensional state spaces, with implications for topological photonics and quantum communications. CONTENTS I. Introduction 1 II. Geometric versus dynamic phase in a nutshell 2 III. The Pancharatnam-Berry phase 2 A. Experimental realisation 2 B. Geometric interpretation 3 IV. The origin of geometric phases 3 A. Fibre bundle theory, a universal model 3 B. From Poincaré to Hopf 4 V. Exploring high dimensional state spaces 5 A. The spin-redirection phase 5 B. Geometric phases of spatial transverse modes 7 C.

Optical control of the complex phase of a quantum ground state amplitude

“…[14]. The geometric phase has a deep connection with the mathematical theory of fibre bundles and holonomies [15], which has also been pursued in polarization optics [16].…”